SUMMARY How cells control their size and maintain size homeostasis is a fundamental open question. Cell-size homeostasis has been discussed in the context of two major paradigms: sizer, in which the cell actively monitors its size and triggers the cell cycle once it reaches a critical size, and timer, in which the cell attempts to grow for a specific amount of time before division. These paradigms, in conjunction with the “growth law” [1] and the quantitative bacterial cell cycle model [2], inspired numerous theoretical models [3-9] and experimental investigations from growth [10,11] to cell cycle and size control [12–15]. However, experimental evidence involved difficult-to-verify assumptions or population-averaged data, which allowed different interpretations [1–5,16–20] or limited conclusions [4–9]. In particular, population-averaged data and correlations are inconclusive as the averaging process masks causal effects at the cellular level. In this work, we extended a microfluidic “mother machine” [21] and monitored hundreds of thousands of Gram-negative Escherichia coli and Gram-positive Bacillus subtilis cells under a wide range of steady-state growth conditions. Our combined experimental results and quantitative analysis demonstrate that cells add a constant volume each generation irrespective of their newborn sizes, conclusively supporting the so-called constant Δ model. This model was introduced for E. coli [6,7] and recently revisited [9], but experimental evidence was limited to correlations. This “adder” principle quantitatively explains experimental data at both the population and single-cell levels, including the origin and the hierarchy of variability in the size-control mechanisms, and how cells maintain size homeostasis.

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A system with many degrees of freedom can be characterized by a covariance matrix; principal components analysis (PCA) focuses on the eigenvalues of this matrix, hoping to find a lower dimensional description. But when the spectrum is nearly continuous, any distinction between components that we keep and those that we ignore becomes arbitrary; it then is natural to ask what happens as we vary this arbitrary cutoff. We argue that this problem is analogous to the momentum shell renormalization group (RG). Following this analogy, we can define relevant and irrelevant operators, where the role of dimensionality is played by properties of the eigenvalue density. These results also suggest an approach to the analysis of real data. As an example, we study neural activity in the vertebrate retina as it responds to naturalistic movies, and find evidence of behavior controlled by a nontrivial fixed point. Applied to financial data, our analysis separates modes dominated by sampling noise from a smaller but still macroscopic number of modes described by a non–Gaussian distribution.